Hearthscience

Overview and Motivation:

Some of our team members are loyal fans of the popular free online card game, Hearthstone: Heroes of Warcraft, which was released worldwide by Blizzard on 2014 with more than 40 million registered Hearthstone accounts by November 2015.

The main element of the game Hearthstone are cards, which consist of a list of features including cost, attack (number of damages can be made to the opponent per turn),health (number of damages that can bear before being destroyed) and other special abilities. Here is an example of the card:

Before every game starts, each of the two players will choose 1 hero mode among the 9 and then select 30 different cards over 700 cards to build his/her own deck depending on the mode. Each turn, the player will draw one card randomly from the 30 cards and one more mana crystal (money). The player can choose the cards to use among all those in hand that cost up to the mana crystals he/she has by that turn. The game ends when one player is attacked to death (lose all 30 units of health) or he/she concedes, and the other player will win.

Therefore, the initial building of the 30 cards, as well as the choices of cards to use during the game will directly influence the results of the game. This motivated us:

• To evaluate the value of all cards.
• To explore the correlation between cards.
• To decide the set of core cards and number of drawing cards.
• To explore the number of different costs of card in a deck with simulations.
• To build a deck with optimal combined strength.

Initial Questions:

1. What are the “true” values of individual cards? Are there any properties the Blizard company used to assign values (cost) of these cards? Is there any card undervalued/overvalued by the company?_

• This was considered as the first stage of our analysis. In this stage, we aimed to estimate the value of each card independently, only considering the basic information from the card itself, i.e. attack, health, special ability.

2. What is the balance between low cost cards and high cost cards?_

• This is the second stage of our analysis. The card a player can use is restricted by the number of mana he/she has for that turn, starting from 1 at the beginning and will reach to a maximum of 10 since the 10th turn. If there is no card with cost less than or equal to the number of mana in that turn, the player cannot use any card. Note that without this consideration, we may choose all 30 cards to be with very high cost – however, we are not able to use them in the first several turns. If the opponent do have small cards, we may be attacked to death before we could ever use any card. We aimed to simulate the possibilities of playing a card in the first 5 turns.

3. Are there any “core” combination of cards?_

• This is the third stage of our analysis. We aimed to estimate the correlation between cards in certain decks of a specific hero and hope to create card combinations based on principal component analyses. In deciding the number of core cards, we also want to explore the number of drawing cards we need, in order to be able to get the core cards in the random drawing during the game.

4. Are we able to build a powerful deck (30 cards) for some heros?_

• With all our analysis in previous stages, we aimed to build potentially powerful decks for certain heros.

5. Test the deck we built (optional)_ * We can test our model by simulating games using the deck and strategy we developed, and calculate its percentage of winning. ## Related Work:

library

Here are the libraries we have used in our project.

library(rjson)
library(dplyr)
library(tidyr)
library(knitr)
library(readr)
library(stringr)
library(ggplot2)
library(gridExtra)
library(graphics)
library(grid)
library(ggrepel)
library(scales)
library(cowplot)
library(rvest)
library(XML)
library(vegan)
library(RColorBrewer)
library(gplots) 
library(devtools)
library(reshape)
library(dendextend)
library(reshape2)
library(VGAM)

Data:

We have two types of data: 1) basic card information (attack/health/cost/description of cards) and 2) frequently used decks from top players.

• Basic card information: We collected card data from Hearthstone’s API: http://hearthstoneapi.com, Card. It contains thorough card information of: card name, card set (Basic, Classic, Blackrock Mountain, Goblins vs Gnomes, Naxxramas, Promotion, The Grand Tournament, The League of Explorers), card type (minions, spells, weapons), player class (Priest, Hunter, Warlock, Shaman, Warrior, Mage, Paladin, Druid, Rogue, All), card description (text), mechanics of minions (Battlecry, Taunt, Normal, HealTarget, Charge, Aura, AdjacentBuff, Spell Damage, Freeze, Enrage, Divine Shield, Stealth, Overload, Deathrattle, Windfury, Combo, Poisonous, Inspire), attack, health, and cost of cards. We have collected the json file using Python and transfer it into RData (and csv files) using R.

## Data wrangling from json to RData:
json_file = "cards2.txt"
data <- fromJSON(file = json_file)
card_category = names(data)

not_empty = which(sapply(1:length(data), function(i){length(data[[i]])})>0)

card_category = card_category[not_empty]

data = lapply(not_empty, function(i){data[[i]]})
data1 = lapply(1:length(data), function(k) {lapply(data[[k]],
                                                   function(i) {lapply(i, function(j){
                                                     j = ifelse(is.null(j),NA,j)})})})

col_names = lapply(1:length(data1),
                   function(k) {
                     lapply(1:length(data1[[k]]),
                            function(i) {names(data1[[k]][[i]])})})

data2 = lapply(1:length(data1), 
           function(k) {
             lapply(1:length(data1[[k]]),
                    function(i) {
                      matrix(unlist(data1[[k]][[i]]), 
                             ncol = length(data1[[k]][[i]]), 
                             byrow = T)})})

for(k in 1:length(data2)){
  colnames(data2[[k]][[1]]) = col_names[[k]][[1]]
  data2[[k]][[1]] = data.frame(data2[[k]][[1]])
  for(i in 2:length(data2[[k]])){
    colnames(data2[[k]][[i]]) = col_names[[k]][[i]]
    data2[[k]][[i]] = data.frame(data2[[k]][[i]])
    data2[[k]][[i]] = bind_rows(data2[[k]][[i-1]],data2[[k]][[i]])
  }
  assign(card_category[k], tbl_df(data2[[k]][[length(data2[[k]])]]))
}

final_data = get(card_category[1])
for (i in 2:length(data2)){
  final_data = bind_rows(final_data, get(card_category[i]))
}
# write.table(final_data, file = "final_data.csv", sep = "\t")
# save(final_data, file = "final_data.RData")

Data wrangling of card descriptions: This part is aimed for detailed classification of minion card descriptions (other than the mechanics they are currently classified as).

1. For some minions that have more than one mechanics (e.g. Taunt that has deathrattle), they are only classified as one of their mechanics in the Hearthstone dataset. The following wrangling aims to classify them with all the mechanics they have with dummy variables (having certain feature = 1 vs. without certain feature = 0).

load("minions_text.RData")
minions_text = tbl_df(minions_text) %>%
  select(-cardId, -flavor, -type, -artist, -collectible, -howToGet, -howToGetGold, -img, -imgGold, -locale, -race, -faction, -elite) %>%
  mutate(playerClass = ifelse(is.na(playerClass), "All", as.character(playerClass)))

minions_text = minions_text %>% 
  mutate(text = as.character(text)) %>%
  mutate(text = gsub("<b>", "", text)) %>%
  mutate(text = gsub("</b>", "", text)) %>%
  mutate(text = gsub("\xa1\xaf", "'", text)) %>%
  mutate(text = ifelse(is.na(text), "None", text)) 

minions_text = minions_text %>%
  mutate(AdjacentBuff= ifelse(text %in% minions_text$text[grep("AdjacentBuff",minions_text$text)], 1, AdjacentBuff))%>% 
   mutate(Aura= ifelse(text %in% minions_text$text[grep("Aura",minions_text$text)], 1, 0))%>% 
   mutate(Battlecry = ifelse(text %in% minions_text$text[grep("Battlecry",minions_text$text)], 1, Battlecry))%>%
  mutate(Charge= ifelse(text %in% minions_text$text[grep("Charge",minions_text$text)], 1, Charge))%>%
 mutate(Combo = ifelse(text %in% minions_text$text[grep("Combo",minions_text$text)], 1, Combo))%>%
  mutate(Deathrattle = ifelse(text %in% minions_text$text[grep("Deathrattle",minions_text$text)], 1, Deathrattle))%>%
  mutate(Divine_Shield = ifelse(text %in% minions_text$text[grep("Divine_Shield",minions_text$text)], 1, Divine_Shield))%>%
  mutate(Enrage = ifelse(text %in% minions_text$text[grep("Enrage",minions_text$text)], 1, Enrage))%>%
  mutate(Inspire = ifelse(text %in% minions_text$text[grep("Inspire",minions_text$text)], 1, Inspire))%>%
  mutate(Overload= ifelse(text %in% minions_text$text[grep("Overload",minions_text$text)], 1, Overload))%>%
  mutate(Poisonous = ifelse(text %in% minions_text$text[grep("Poisonous",minions_text$text)], 1, Poisonous))%>%
  mutate(Windfury = ifelse(text %in% minions_text$text[grep("Windfury",minions_text$text)], 1, Windfury))

2. The beauty of Hearthstone (and the most difficult part for quantitative analysis) is that almost every minion has its unique feature that are described in text on the card. Therefore, it’s hard to “value” a card without taking these descriptions into account while on the other hand, texts them selves are difficult to be simply quantified. The following wrangling aims to identify certain verbs (deal, restore, etc.) and nouns (attacks, healths, etc.) frequently used in the card description and tried to classifiy cards with more features that were not classified by their mechanics.

minions_text = minions_text %>%
  mutate(Choice = ifelse(text %in% minions_text$text[grep("; or",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep("if",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep("whenever",minions_text$text, ignore.case = T)], 1, Conditional)) %>% 
  mutate(Conditional = ifelse(text %in% minions_text$text[grep(",",minions_text$text, ignore.case = T)], 1, Conditional)) %>% 
  mutate(Add = ifelse(text %in% minions_text$text[grep("add",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Cast = ifelse(text %in% minions_text$text[grep("cast",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Deal = ifelse(text %in% minions_text$text[grep("Deal",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Destroy = ifelse(text %in% minions_text$text[grep("destroy",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Discover = ifelse(text %in% minions_text$text[grep("discover",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Draw = ifelse(text %in% minions_text$text[grep("draw",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Discard = ifelse(text %in% minions_text$text[grep("discard",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Freeze = ifelse(text %in% minions_text$text[grep("freeze",minions_text$text, ignore.case = T)], 1, Freeze)) %>% 
  mutate(Gain = ifelse(text %in% minions_text$text[grep("gain",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Give = ifelse(text %in% minions_text$text[grep("give",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Reduce = ifelse(text %in% minions_text$text[grep("reduce",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Remove = ifelse(text %in% minions_text$text[grep("remove",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Restore = ifelse(text %in% minions_text$text[grep("restore",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Reveal = ifelse(text %in% minions_text$text[grep("reveal",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Silence = ifelse(text %in% minions_text$text[grep("silence",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Summon = ifelse(text %in% minions_text$text[grep("summon",minions_text$text, ignore.case = T)], 1, 0)) %>% 
  mutate(Trigger = ifelse(text %in% minions_text$text[grep("trigger",minions_text$text, ignore.case = T)],1,0)) %>%
  mutate(Number_within = ifelse(text %in% minions_text$text[grep("+[0-9]", minions_text$text)],1,0))%>%
  mutate(Attack = ifelse(text %in% minions_text$text[grep("attack",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Health = ifelse(text %in% minions_text$text[grep("health",minions_text$text, ignore.case = T)], 1, 0))%>%
  mutate(Damage = ifelse(text %in% minions_text$text[grep("damage",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Cant = ifelse(text %in% minions_text$text[grep("can't",minions_text$text, ignore.case = T)], 1, 0)) %>%
  mutate(Nothing = ifelse(text == "None", 1, 0))

colnames(minions_text)

save(minions_text, file = "minions_text.RData")

Exploratory Analysis

theme_set(theme_bw(base_size = 16))

load("minions_text.RData")
data<-minions_text

distribution of Cost

#remove costs that are "12" and "20" for these two cards are very special
Cost<-data%>%dplyr::arrange(cost)
Cost<-unique(data%>%filter(cost<=10)%>%group_by(cost)%>%mutate(n=n())%>%ungroup()%>%select(cost,n))
# 7 stands for higher than 7
Cost1<-Cost%>%filter(cost<7)                       
Cost2<-c(7,61)
Cost<-rbind(Cost1,Cost2)

Cost<-Cost%>%mutate(pos=cumsum(n)-n/2)
p<-Cost%>%ggplot(aes(x=1,y=n,fill=factor(cost)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of Card Cost")

#histogram
qplot(data$cost,geom="histogram",xlab="cost",main="Histogram for cost",binwidth=1)

Conclusion: cards with cost “2”,“3”,“4” out of the 11 possible costs occupying around 54% in total are most common in the deck

distribution of attack

Attackk<-data%>%arrange(attack)
Attack<-unique(Attackk%>%group_by(attack)%>%mutate(n=n())%>%ungroup()%>%select(attack,n))
#histogram
qplot(data$attack,geom="histogram",xlab="attack",main="Histogram for attack",binwidth=1)

distribution of health

Health<-data%>%arrange(health)
Health<-unique(Health%>%group_by(health)%>%mutate(n=n())%>%ungroup()%>%select(health,n))
#histogram
qplot(data$health,geom="histogram",xlab="health",main="Histogram for health",binwidth=1)

distribution of mechanics

Mechanics<-data%>%arrange(mechanics)
Mechanics<-unique(Mechanics%>%group_by(mechanics)%>%mutate(n=n())%>%ungroup()%>%select(mechanics,n))
#histogram
qplot(data$mechanics,xlab="mechanics",main="Histogram for Mechanics")+theme(axis.text.x = element_text(angle = 45, hjust = 1))

*distributions for Charge

charge<-data%>%filter(mechanics=="Charge")
Cost<-charge%>%dplyr::arrange(cost)
Cost<-unique(charge%>%filter(cost<=10)%>%group_by(cost)%>%mutate(n=n())%>%ungroup()%>%select(cost,n))
Cost<-Cost%>%mutate(pos=cumsum(n)-n/2)
p<-Cost%>%ggplot(aes(x=1,y=n,fill=factor(cost)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of Card Cost for Charge")

Attackk<-charge%>%arrange(attack)
Attack<-unique(charge%>%group_by(attack)%>%mutate(n=n())%>%ungroup()%>%select(attack,n))
Health<-charge%>%arrange(health)
Health<-unique(Health%>%group_by(health)%>%mutate(n=n())%>%ungroup()%>%select(health,n))
#histogram
qplot(charge$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Charge",binwidth=1)

qplot(charge$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Charge",binwidth=1)

qplot(charge$health,geom="histogram",xlab="health",main="Histogram of health distribution for Charge",binwidth=1)

*distributions for Divine Shield

ds<-data%>%filter(mechanics=="Divine Shield")
Cost<-ds%>%dplyr::arrange(cost)
Cost<-unique(ds%>%filter(cost<=10)%>%group_by(cost)%>%mutate(n=n())%>%ungroup()%>%select(cost,n))
Cost<-Cost%>%mutate(pos=cumsum(n)-n/2)
p<-Cost%>%ggplot(aes(x=1,y=n,fill=factor(cost)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of Card Cost for Divine Shield")

Attackk<-ds%>%arrange(attack)
Attack<-unique(ds%>%group_by(attack)%>%mutate(n=n())%>%ungroup()%>%select(attack,n))
Health<-ds%>%arrange(health)
Health<-unique(Health%>%group_by(health)%>%mutate(n=n())%>%ungroup()%>%select(health,n))
#histogram
qplot(ds$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Divine Shield",binwidth=1)

qplot(ds$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Divine Shield",binwidth=1)

qplot(ds$health,geom="histogram",xlab="health",main="Histogram of health distribution for Divine Shield",binwidth=1)

*distributions for Taunt

Taunt<-data%>%filter(mechanics=="Taunt")
Cost<-Taunt%>%dplyr::arrange(cost)
Cost<-unique(Taunt%>%filter(cost<=10)%>%group_by(cost)%>%mutate(n=n())%>%ungroup()%>%select(cost,n))
Cost<-Cost%>%mutate(pos=cumsum(n)-n/2)
p<-Cost%>%ggplot(aes(x=1,y=n,fill=factor(cost)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of Card Cost for Taunt")

Attackk<-Taunt%>%arrange(attack)
Attack<-unique(Taunt%>%group_by(attack)%>%mutate(n=n())%>%ungroup()%>%select(attack,n))
Health<-Taunt%>%arrange(health)
Health<-unique(Health%>%group_by(health)%>%mutate(n=n())%>%ungroup()%>%select(health,n))
#histogram
p1<-qplot(Taunt$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Divine Shield",binwidth=1)
p2<-qplot(Taunt$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Divine Shield",binwidth=1)
qplot(Taunt$health,geom="histogram",xlab="health",main="Histogram of health distribution for Divine Shield",binwidth=1)

distribution of cardSet

cs<-data%>%arrange(cardSet)
cs<-unique(cs%>%group_by(cardSet)%>%mutate(n=n())%>%ungroup()%>%select(cardSet,n))
#pie chart
cs<-cs%>%mutate(pos=cumsum(n)-n/2)
p<-cs%>%ggplot(aes(x=1,y=n,fill=factor(cardSet)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of cardSet")

#histogram
qplot(data$cardSet,xlab="cardSet",main="Histogram for CardSet distribution")+theme(axis.text.x = element_text(angle = 45, hjust = 1))

###basic carset
basic<-data%>%filter(cardSet=="Basic")
qplot(basic$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for basic cardset",binwidth=1)

qplot(basic$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for basic cardset",binwidth=1)

qplot(basic$health,geom="histogram",xlab="health",main="Histogram of health distribution for basic cardset",binwidth=1)

###Blackrock Mountain
bm<-data%>%filter(cardSet=="Blackrock Mountain")
qplot(bm$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Blackrock Mountain",binwidth=1)

qplot(bm$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Blackrock Mountain",binwidth=1)

qplot(bm$health,geom="histogram",xlab="health",main="Histogram of health distribution for Blackrock Mountain",binwidth=1)

###Classic
Classic<-data%>%filter(cardSet=="Classic")
qplot(Classic$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Classic",binwidth=1)

qplot(Classic$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Classic",binwidth=1)

qplot(Classic$health,geom="histogram",xlab="health",main="Histogram of health distribution for Classic",binwidth=1)

###Goblins vs Gnomes
gg<-data%>%filter(cardSet=="Goblins vs Gnomes")
qplot(gg$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Goblins vs Gnomes",binwidth=1)

qplot(gg$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Goblins vs Gnomes",binwidth=1)

qplot(gg$health,geom="histogram",xlab="health",main="Histogram of health distribution for Goblins vs Gnomes",binwidth=1)

###Naxxramas
na<-data%>%filter(cardSet=="Naxxramas")
qplot(na$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Naxxramas",binwidth=1)

qplot(na$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Naxxramas",binwidth=1)

qplot(na$health,geom="histogram",xlab="health",main="Histogram of health distribution for Naxxramas",binwidth=1)

###Promotion
Promotion<-data%>%filter(cardSet=="Naxxramas")
qplot(Promotion$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Promotion",binwidth=1)

qplot(Promotion$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Promotion",binwidth=1)

qplot(Promotion$health,geom="histogram",xlab="health",main="Histogram of health distribution for Promotion",binwidth=1)

###The Grand Tournament
tgt<-data%>%filter(cardSet=="The Grand Tournament")
qplot(tgt$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for The Grand Tournament",binwidth=1)

qplot(tgt$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for The Grand Tournament",binwidth=1)

qplot(tgt$health,geom="histogram",xlab="health",main="Histogram of health distribution for The Grand Tournament",binwidth=1)

###The League of Explorers
tloe<-data%>%filter(cardSet=="The League of Explorers")
qplot(tloe$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for The League of Explorers",binwidth=1)

qplot(tloe$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for The League of Explorers",binwidth=1)

qplot(tloe$health,geom="histogram",xlab="health",main="Histogram of health distribution for The League of Explorers",binwidth=1)

distribution of rarity

rr<-unique(data%>%group_by(rarity)%>%mutate(n=n())%>%ungroup()%>%select(rarity,n))
#pie chart
rr<-rr%>%mutate(pos=cumsum(n)-n/2)
p<-rr%>%ggplot(aes(x=1,y=n,fill=factor(rarity)))
p+geom_bar(stat="identity",width=1)+geom_text(aes(x=1.6,y=pos,label = percent(n/sum(n))))+coord_polar(theta="y")+ xlab('')+ylab('')+theme(axis.text=element_blank(),axis.ticks=element_blank(),panel.grid=element_blank())+ggtitle("Pie Chart of rarity")

#histogram
qplot(data$rarity,xlab="rarity",main="Histogram for rarity")+theme(axis.text.x = element_text(angle = 45, hjust = 1))

###Common
common<-data%>%filter(rarity=="Common")
qplot(common$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for common",binwidth=1)

qplot(common$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for common",binwidth=1)

qplot(common$health,geom="histogram",xlab="health",main="Histogram of health distribution for common",binwidth=1)

###Epic
Epic<-data%>%filter(rarity=="Epic")
qplot(Epic$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Epic",binwidth=1)

qplot(Epic$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Epic",binwidth=1)

qplot(Epic$health,geom="histogram",xlab="health",main="Histogram of health distribution for Epic",binwidth=1)

###Free
Free<-data%>%filter(rarity=="Free")
qplot(Free$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Free",binwidth=1)

qplot(Free$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Free",binwidth=1)

qplot(Free$health,geom="histogram",xlab="health",main="Histogram of health distribution for Free",binwidth=1)

###Legendary
Legendary<-data%>%filter(rarity=="Legendary")
qplot(Legendary$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Legendary",binwidth=1)

qplot(Legendary$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Legendary",binwidth=1)

qplot(Legendary$health,geom="histogram",xlab="health",main="Histogram of health distribution for Legendary",binwidth=1)

###Rare
Rare<-data%>%filter(rarity=="Rare")
qplot(Rare$cost,geom="histogram",xlab="cost",main="Histogram of cost distribution for Rare",binwidth=1)

qplot(Rare$attack,geom="histogram",xlab="attack",main="Histogram of attack distribution for Rare",binwidth=1)

qplot(Rare$health,geom="histogram",xlab="health",main="Histogram of health distribution for Rare",binwidth=1)

Final Analysis:

Card Value Analysis

1. What are the “true” values of individual cards? Are there any properties the Blizard company used to assign values (cost) of these cards? Is there any card undervalued/overvalued by the company?

load("minions_text.RData")
## cost vs attack+health:
minions_text %>% ggplot(aes(cost)) + stat_bin(aes(y = ..count..), bins = 50 , position='dodge')

minions_text %>% mutate(attplusheal = attack+health) %>% ggplot(aes(attplusheal)) + stat_bin(aes(y = ..count..), bins = 50 , position='dodge')

From the above plots, we can found similar distributions between the cost and the sum of attach and health, where the distributions are right-skewed. Also, there seems to be some outliers that are very different from other cards.

minions_text %>%
  filter(cost > 10) %>%
  select(name, cost, attack, health, mechanics, playerClass)

## Source: local data frame [3 x 6]
## 
##              name  cost attack health mechanics playerClass
##            (fctr) (int)  (int)  (int)     (chr)       (chr)
## 1  Mountain Giant    12      8      8    Normal         All
## 2    Molten Giant    20      8      8    Normal         All
## 3 Clockwork Giant    12      8      8    Normal         All

It might be a good idea to filter out these cards.

minions_text = minions_text %>% mutate(attplusheal = attack+health) %>% filter(cost <= 10)

## cost vs attack+health:
minions_text %>% mutate(attplusheal = attack+health) %>% 
  group_by(attplusheal) %>%
  summarize(cost = mean(cost)) %>%
  ggplot(aes(attplusheal, cost)) + geom_point()

We can see from the above graph that higher attplusheal value (attack+health) is associated with higher mean cost.

In Hearthstone, the cost of cards is usually categorized into 0 ~ 6 and 7+. Here, we wrangled the card costs into these 8 categories and also separate them by cardSet:

## All:
minions_text = minions_text %>% 
  mutate(cost1 = ifelse(cost >= 7, 7, cost))
minions_text %>% ggplot(aes(cost1)) + geom_histogram()

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

minions_text %>% mutate(attplusheal = attack+health) %>% 
  group_by(cost1, attplusheal) %>% summarize(count = n()) %>%
  ggplot(aes(attplusheal, cost1, col = factor(floor(count/10)*10))) + geom_point()

## by cardSet:
minions_text %>% ggplot(aes(cost1, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 20 , position='dodge')

## by attack, cost, health:
minions_bars = minions_text %>% gather(key, value, cost, attack, health)
minions_bars %>% ggplot(aes(value, group = key, fill = key)) + stat_bin(aes(y = ..count..), bins = 40, position='dodge')

## by cardSet:
## Cost:
minions_text %>% ggplot(aes(cost, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

## Attack:
minions_text %>% ggplot(aes(attack, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

## Health:
minions_text %>% ggplot(aes(health, group = cardSet, fill = cardSet)) + stat_bin(aes(y = ..count..), bins = 40 , position='dodge')

Since the outcome variable (Y) in our analysis is the costs of cards, which are normally integer from 0 to 7+ (all values greater than 7 are considered in the group of 7+), we adopted a model that consider ordinal polytomous outcome – cumulative logits model. Since the features’ effects (attack, cost, special abilities, etc.) should be the similar in cards with different costs, we also assumed proportional odds of these features across different cost groups. And we ended up with 7 outcome groups (cost value: 1 to 7+), we excluded cards that cost 0 mana since 1) they are usually cards that do not cost players to play and 2) the nature of these 0 cost cards are quite different from normal minion cards. In general, the cumulative logits model is in format shown below, where X is the covariate matrix, and \(\beta\) is the coefficient matrix:

\[\mbox{logit(Pr}{(Y \leq k | X_i = x_i))} = \beta_{k0} + \sum \beta_{ki}*x_i\]

Using this cumulative logits model, we are able to estimate the probability of a card being classified in each cost group (p1 to p7), and then by conditioning on the features of a card, we are able to assign a value of that card with the maximum probability among p1 to p7 (the most likely cost of a card based on its features).

Since one of our assumption that the cost of a card is proportional to the damage it can lead to, we first considered a univariate model which include attack as the only covariate:

## X: attack
## Y: cost

minions_text1 = minions_text %>% 
  filter(cost != 0) %>%
  arrange(cost) %>%
  mutate(Y1 = ifelse(cost == 1, 1, 0)) %>%
  mutate(Y2 = ifelse(cost == 2, 1, 0)) %>%
  mutate(Y3 = ifelse(cost == 3, 1, 0)) %>%
  mutate(Y4 = ifelse(cost == 4, 1, 0)) %>%
  mutate(Y5 = ifelse(cost == 5, 1, 0)) %>%
  mutate(Y6 = ifelse(cost == 6, 1, 0)) %>%
  mutate(Y7 = ifelse(cost >= 7, 1, 0)) 

set.seed(1001)
n_test <- round(nrow(minions_text1) / 10)
test_indices <- sample(1:nrow(minions_text1), n_test, replace=FALSE)
test <- minions_text1[test_indices,]
train <- minions_text1[-test_indices,]

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ attack, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack:
test1 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*attack)/(1+exp(coef1[1,]+coef1[2,]*attack)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*attack)/(1+exp(coef2[1,]+coef2[2,]*attack))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*attack)/(1+exp(coef3[1,]+coef3[2,]*attack))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*attack)/(1+exp(coef4[1,]+coef4[2,]*attack))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*attack)/(1+exp(coef5[1,]+coef5[2,]*attack))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*attack)/(1+exp(coef6[1,]+coef6[2,]*attack))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test1 = test %>% left_join(test1, by = "cardId")
RMSE <- function(true_ratings, predicted_ratings){
    sqrt(mean((true_ratings - predicted_ratings)^2))
}
model1 = RMSE(test1$cost1, test1$value)
rmse_results = data_frame(method = "X: attack", RMSE = model1)

Since the cost of a card can also be influenced by the time it can survive on the stage, we also included some potential effect of health by summing up both attack and health (attack+health ) as a univariate:

## X: attplusheal
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ attplusheal, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack plus health:
test2 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*attplusheal)/(1+exp(coef1[1,]+coef1[2,]*attplusheal)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*attplusheal)/(1+exp(coef2[1,]+coef2[2,]*attplusheal))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*attplusheal)/(1+exp(coef3[1,]+coef3[2,]*attplusheal))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*attplusheal)/(1+exp(coef4[1,]+coef4[2,]*attplusheal))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*attplusheal)/(1+exp(coef5[1,]+coef5[2,]*attplusheal))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*attplusheal)/(1+exp(coef6[1,]+coef6[2,]*attplusheal))) - p1 - p2 - p3 - p4 - p5) %>% 
  mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test2 = test %>% left_join(test2, by = "cardId")
model2 = RMSE(test2$cost1, test2$value)
rmse_results =  bind_rows(rmse_results, data_frame(method = "X: attplusheal", RMSE = model2))

It seemed like the univariate attack+health worked well in the model, since my testing the model in our testing set, the RMSE decreased. Also, we considered a model which include attack and health separately:

## X: attack, health
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack and health:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>% left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [3 x 2]
## 
##              method      RMSE
##               (chr)     (dbl)
## 1         X: attack 1.0973065
## 2    X: attplusheal 0.8451543
## 3 X: attack, health 0.8451543

We then considered a model which include some feature of the cards, and Charge and Overload were the only features that were significant:

## X: attack, health, charge
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack, health, and charge:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, cardId, playerClass, mechanics) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [4 x 2]
## 
##                      method      RMSE
##                       (chr)     (dbl)
## 1                 X: attack 1.0973065
## 2            X: attplusheal 0.8451543
## 3         X: attack, health 0.8451543
## 4 X: attack, health, charge 0.8806306

## X: attack, health, charge, and overload
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Overload, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}


# To estimate the cost of cards based on attack, health, charge and overload:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, cardId, playerClass, mechanics) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge, overload",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [5 x 2]
## 
##                                method      RMSE
##                                 (chr)     (dbl)
## 1                           X: attack 1.0973065
## 2                      X: attplusheal 0.8451543
## 3                   X: attack, health 0.8451543
## 4           X: attack, health, charge 0.8806306
## 5 X: attack, health, charge, overload 0.8921426

## X: attack, health, charge, divine shield, taunt
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Divine_Shield + Taunt, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack, health, charge, divine shield, and taunt:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, mechanics, name, cardId) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge, divine shield, taunt",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [6 x 2]
## 
##                                            method      RMSE
##                                             (chr)     (dbl)
## 1                                       X: attack 1.0973065
## 2                                  X: attplusheal 0.8451543
## 3                               X: attack, health 0.8451543
## 4                       X: attack, health, charge 0.8806306
## 5             X: attack, health, charge, overload 0.8921426
## 6 X: attack, health, charge, divine shield, taunt 0.8806306

## X: attack, health, charge, overload
## Y: cost

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Overload, cumulative(parallel = T, reverse = F), data = train)
# summary(fitCL)


for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack, health, charge, overload:
test3 = test %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

test3 = test %>%
  select(cost, cost1, attack, health, mechanics, name, cardId) %>% 
  left_join(test3, by = "cardId")
model3 = RMSE(test3$cost1, test3$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge, overload",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [7 x 2]
## 
##                                            method      RMSE
##                                             (chr)     (dbl)
## 1                                       X: attack 1.0973065
## 2                                  X: attplusheal 0.8451543
## 3                               X: attack, health 0.8451543
## 4                       X: attack, health, charge 0.8806306
## 5             X: attack, health, charge, overload 0.8921426
## 6 X: attack, health, charge, divine shield, taunt 0.8806306
## 7             X: attack, health, charge, overload 0.8921426

## X: attack, health, charge, divine shield, taunt in warlock
## Y: cost

train_warlock = train %>% filter(playerClass == "All" | playerClass == "Warlock")
fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Divine_Shield + Taunt, cumulative(parallel = T, reverse = F), data = train_warlock)
# summary(fitCL)

test_warlock = test %>% filter(playerClass == "All" | playerClass == "Warlock")

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}

# To estimate the cost of cards based on attack, health, charge, divine shield, and taunt in warlock:
test_warlock = test_warlock %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId, cost1, attack, health) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

model3 = RMSE(test_warlock$cost1, test_warlock$value)
rmse_results = bind_rows(rmse_results,data_frame(method="X: attack, health, charge, divine shield, taunt in warlock",  
                                     RMSE = model3))
rmse_results

## Source: local data frame [8 x 2]
## 
##                                                       method      RMSE
##                                                        (chr)     (dbl)
## 1                                                  X: attack 1.0973065
## 2                                             X: attplusheal 0.8451543
## 3                                          X: attack, health 0.8451543
## 4                                  X: attack, health, charge 0.8806306
## 5                        X: attack, health, charge, overload 0.8921426
## 6            X: attack, health, charge, divine shield, taunt 0.8806306
## 7                        X: attack, health, charge, overload 0.8921426
## 8 X: attack, health, charge, divine shield, taunt in warlock 0.9393364

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Divine_Shield + Taunt, cumulative(parallel = T, reverse = F), data = minions_text1)
summary(fitCL)

## 
## Call:
## vglm(formula = cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + 
##     Charge + Divine_Shield + Taunt, family = cumulative(parallel = T, 
##     reverse = F), data = minions_text1)
## 
## Pearson residuals:
##                    Min        1Q    Median        3Q    Max
## logit(P[Y<=1])  -1.595 -0.166733 -0.046307 -0.009202  5.593
## logit(P[Y<=2])  -2.922 -0.219257 -0.047367  0.196809  9.030
## logit(P[Y<=3]) -13.386 -0.207826 -0.005203  0.235558  9.888
## logit(P[Y<=4])  -6.959 -0.096409  0.057904  0.179912 19.107
## logit(P[Y<=5])  -5.325  0.008591  0.040716  0.147200 22.079
## logit(P[Y<=6]) -29.253  0.008756  0.025017  0.083758 20.022
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept):1  2.87739    0.29498   9.754  < 2e-16 ***
## (Intercept):2  5.17179    0.32551  15.888  < 2e-16 ***
## (Intercept):3  7.14499    0.39177  18.238  < 2e-16 ***
## (Intercept):4  8.98798    0.46589  19.292  < 2e-16 ***
## (Intercept):5 10.81457    0.54491  19.847  < 2e-16 ***
## (Intercept):6 13.06520    0.66349  19.692  < 2e-16 ***
## health        -1.05901    0.07188 -14.734  < 2e-16 ***
## attack        -1.04702    0.07206 -14.530  < 2e-16 ***
## Charge        -1.25559    0.41765  -3.006  0.00264 ** 
## Divine_Shield -1.24240    0.77257  -1.608  0.10781    
## Taunt          0.32869    0.36830   0.892  0.37215    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Number of linear predictors:  6 
## 
## Dispersion Parameter for cumulative family:   1
## 
## Residual deviance: 1151.674 on 2917 degrees of freedom
## 
## Log-likelihood: -575.8368 on 2917 degrees of freedom
## 
## Number of iterations: 7 
## 
## Exponentiated coefficients:
##        health        attack        Charge Divine_Shield         Taunt 
##     0.3467998     0.3509830     0.2849067     0.2886917     1.3891540

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}
final = minions_text %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Divine_Shield+coef1[6,]*Taunt))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId, cost1, attack, health, name, playerClass, mechanics) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

final %>% filter(value != cost1) %>%
  mutate(resid = value - cost1) %>%
  ggplot(aes(resid, group = mechanics, fill = mechanics)) + stat_bin(aes(y = ..count..), bins = 10 , position='dodge') 

fitCL = vglm(cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + Charge + Overload, cumulative(parallel = T, reverse = F), data = minions_text1)
summary(fitCL)

## 
## Call:
## vglm(formula = cbind(Y1, Y2, Y3, Y4, Y5, Y6, Y7) ~ health + attack + 
##     Charge + Overload, family = cumulative(parallel = T, reverse = F), 
##     data = minions_text1)
## 
## Pearson residuals:
##                    Min        1Q    Median        3Q    Max
## logit(P[Y<=1])  -1.575 -0.186459 -0.045147 -0.008935  5.714
## logit(P[Y<=2])  -2.904 -0.216306 -0.046015  0.161136  9.266
## logit(P[Y<=3]) -13.310 -0.203616 -0.004975  0.237888 10.145
## logit(P[Y<=4])  -6.911 -0.094560  0.060071  0.180946 19.666
## logit(P[Y<=5])  -5.402  0.008470  0.040499  0.149701 22.612
## logit(P[Y<=6]) -28.977  0.008744  0.025126  0.084399 20.890
## 
## Coefficients:
##               Estimate Std. Error z value Pr(>|z|)    
## (Intercept):1  2.86361    0.29233   9.796  < 2e-16 ***
## (Intercept):2  5.16869    0.32353  15.976  < 2e-16 ***
## (Intercept):3  7.14835    0.39052  18.305  < 2e-16 ***
## (Intercept):4  9.00975    0.46637  19.319  < 2e-16 ***
## (Intercept):5 10.85696    0.54705  19.846  < 2e-16 ***
## (Intercept):6 13.08782    0.66221  19.764  < 2e-16 ***
## health        -1.06173    0.07185 -14.777  < 2e-16 ***
## attack        -1.05602    0.07216 -14.635  < 2e-16 ***
## Charge        -1.23041    0.41724  -2.949  0.00319 ** 
## Overload       1.85670    0.65451   2.837  0.00456 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Number of linear predictors:  6 
## 
## Dispersion Parameter for cumulative family:   1
## 
## Residual deviance: 1147.647 on 2918 degrees of freedom
## 
## Log-likelihood: -573.8236 on 2918 degrees of freedom
## 
## Number of iterations: 7 
## 
## Exponentiated coefficients:
##    health    attack    Charge  Overload 
## 0.3458580 0.3478377 0.2921725 6.4025641

for(i in 1: 6){
  assign(paste("coef",i, sep = ""), as.data.frame((coef(fitCL, matrix = T)[,i])))
}
final = minions_text %>% mutate(p1 = as.numeric(exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef1[1,]+coef1[2,]*health+coef1[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)))) %>%
  mutate(p2 = as.numeric(exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef2[1,]+coef2[2,]*health+coef2[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1) %>%
  mutate(p3 = as.numeric(exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef3[1,]+coef3[2,]*health+coef3[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2) %>%
  mutate(p4 = as.numeric(exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef4[1,]+coef4[2,]*health+coef4[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3) %>%
  mutate(p5 = as.numeric(exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef5[1,]+coef5[2,]*health+coef5[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4) %>%
  mutate(p6 = as.numeric(exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload)/(1+exp(coef6[1,]+coef6[2,]*health+coef6[3,]*attack+coef1[4,]*Charge+coef1[5,]*Overload))) - p1 - p2 - p3 - p4 - p5) %>% mutate(p7 = 1 - p1 - p2 - p3 - p4 - p5 - p6) %>% 
  mutate(value = 7) %>% 
  group_by(cardId, cost1, attack, health, name, playerClass, mechanics) %>%
  summarize(value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p1, 1, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p2, 2, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p3, 3, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p4, 4, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p5, 5, value),
              value = ifelse(max(p1,p2,p3,p4,p5,p6,p7) == p6, 6, value))

final %>% filter(value != cost1) %>%
  mutate(resid = value - cost1) %>%
  ggplot(aes(resid, group = mechanics, fill = mechanics)) + stat_bin(aes(y = ..count..), bins = 10 , position='dodge') 

Simulation

2. What is the balance between small cost cards and big cost cards? Assumptions: 1. Players will not use the card with cost 0 in the earlier several turns. 2. Cost can roughly represent the value of the card, thus we can maximum the cost of all 30 cards to maximum the value. 3. We focus on the first 5 turns.

First, create decks with all reasonable combinations of small cards (1-5) and others.

decks <- expand.grid(n1=0:6, n2=0:6, n3=0:6, n4=0:6, n5=0:6)
decks <- decks %>% tbl_df %>% mutate(others = 30-n1-n2-n3-n4-n5)

Next, use similation to estimate the probability to use card in the first 1/2/3/4/5-turn for each deck. Estimations are made for offensive player, as the defensive player has higher possiblity to use cards (4 cards at the begining with a special 0 cost card that temporatily increases the mana by 1) for the first few turns.

prob_usecard <- function(deck){
        card <- rep(c(1,2,3,4,5,10), deck)
                
        # offensive player
        temp <- t(replicate(1000,sample(card,30)))
        # assume choosing the 3 smallest cards for the starting hand
        sortcard <- t(apply(temp[,1:6],1,sort))
        temp[,1:6] <- sortcard
        sortcard2 <- t(apply(temp[,4:30],1,function(x){sample(x,27)}))
        temp[,4:30] <- sortcard2
        rm(sortcard)
        rm(sortcard2)
        
        # p1: can use card in the first turn
        p1 <- mean(apply(temp[,1:4],1,function(c){as.numeric(sum(c<2)>0)}))  
        # p2: can use card in the first 2 turns
        p2 <- mean(apply(temp[,1:5],1,function(c){as.numeric(sum(c<3)>0)}))  
        # p3: can use card in the first 3 turns
        p3 <- mean(apply(temp[,1:6],1,function(c){as.numeric(sum(c<4)>0)}))  
        # p4: can use card in the first 4 turns
        p4 <- mean(apply(temp[,1:7],1,function(c){as.numeric(sum(c<5)>0)}))  
        # p5: can use card in the first 5 turns
        p5 <- mean(apply(temp[,1:8],1,function(c){as.numeric(sum(c<6)>0)}))  
        
        c(p1, p2, p3, p4, p5)     
}

# get the probability of using card and combine
usecard <- t(apply(decks,1,prob_usecard))
colnames(usecard) <- c("p1","p2","p3","p4","p5")
decks <- cbind(decks,usecard) %>% 
        # add the total cost for each deck
        mutate(sum = n1+2*n2+3*n3+4*n4+5*n5+10*others) 
rm(usecard)

# save simulation results 
write.csv(decks,file="/Users/Yinnan/Desktop/2016/HearthScience/simulation.csv")

# get the simulation result from github
url <- "https://raw.githubusercontent.com/jihua0125/HearthScience/master/simulation.csv"
decks <- read_csv(url)

decks <- decks[,-1]

# constrain on probability of using card
decks.constrain <- decks %>% tbl_df %>% filter(p4>0.95, p2>0.5, p3>0.9, others>10) %>%
        arrange(desc(sum)) 

decks.constrain %>% summarize(min2 = min(n1+n2), min3 = min(n1+n2+n3), min4 = min(n1+n2+n3+n4))

## Source: local data frame [1 x 3]
## 
##    min2  min3  min4
##   (int) (int) (int)
## 1     2     6     7